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Questions tagged [statistical-mechanics]

The study of large, complicated systems by means of statistics and probability theory, in order to extract average properties and to provide a connection between mechanics and thermodynamics.

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In a ball with random thread/strings, how does the density of threads/strings change with radius?

A large plastic ball full of holes is given. (So the holes are in a plastic shell.) Straight threads connect these holes randomly, by passing through the interior of the ball/shell. For a big ball or ...
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1answer
72 views

Why does entropy of an ideal gas break the third law of thermodynamics? [duplicate]

I was recently looking at the entropy equation for an ideal gas, show below $$S=NR\left\{\frac{5}{2}+\ln\left[\frac{V}{N}\left(\frac{2\pi mRT}{h^2}\right)^{3/2}\right]\right\}$$ And what troubles me ...
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How do irrational numbers give incommensurate potential (in lattice models)?

I am trying to understand Aubry-Andre model. It has the following form $$H = \sum_n c_n^\dagger c_{n+1}+H.C.+V\sum_n \cos{(2\pi\beta n)}c_n^\dagger c_n$$ This reference (at 3rd page) says that if $\...
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1answer
83 views

Partition Function of an Ideal Gas

Which is the correct partition function for an ideal (bosonic) gas at high $T$: 1) Sum over the number of particles in each momentum state: $$ z_{\vec{p}} = 1 + e^{- \varepsilon_{\vec{p}}/T} + ... = ...
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1answer
76 views

Problem with indistinguishability in partition function

Consider an ideal gas of classical particles of mass $m$ in uniform potential $\xi$ in 3d. The gas $N$ molecules, volume $V$ and is at temperature $T$. I believe that the Hamiltonian of this system is ...
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First and second order phase transitions

Recently I've been puzzling over the definitions of first and second order phase transitions. The Wikipedia article starts by explaining that Ehrenfest's original definition was that a first-order ...
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+50

Does the Bohr van Leeuwen Theorem also apply to ferromagnetism?

I know that the Bohr-van Leeuwen theorem shows that there could be not consistent pure classical explanation of dia- and paramagnetism. Does the same theorem also rule out a consistent classical ...
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1answer
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Do Boltzmann Mechanics imply we're the only example of sentience in our observable universe?

My understanding of BM is that the Big Bang was just a statistical event. Under this model, many Big Bangs occur which result in a 2-power distribution of universes; there are twice as many universes ...
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0answers
29 views

Does thermal equilibration affect compression work for an ideal gas?

Consider compression of an ideal gas that is uniform in pressure but nonuniform in temperature, from volume $V_1$ to $V_2$. For simplicity of the problem let's say the gas is divided into 2 regions $A$...
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0answers
17 views

Maximizing mutual information with Restricted Boltzmann Machines and Monte-Carlo sampling

So I've been reading through Koch-Janusz and Ringel's, "Mutual Information, Neural Networks, and the Renormalization Group" (check it out here). I'm currently trying to reimplement some results from ...
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1answer
120 views

Density function in phase space

What does density function in phase space physically mean? How does it indicate, the more familiar density that we are accustomed to ( an analogy may be), in phase space?
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Is the second law of thermodynamics a “no-go” theorem?

As defined here, there are several no-go theorems in theoretical physics. These theorems are statements of impossibility. The second law of thermodynamics may be stated in several ways, some of which ...
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1answer
91 views

Two definitions of a macrostate

I fully understand the concept of a microstate, but I'm having some trouble reconciling two seemingly contradictory definitions of a macrostate. Here are the two definitions: A macrostate is defined ...
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0answers
27 views

Tunneling in quantum mechanics and domain wall in 1d Ising model

I am following David Tong's lecture notes on Statistical Field Theory. You can find it here. In page 51-52, he said the domain wall in 1d Ising model is the same as quantum tunnelling in quantum ...
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0answers
19 views

Partition function for a system of particles in a lattice

The problem A $d$ dimensional lattice system with $N$ sites is occupied by particles ($n_i$ particles at site $i$) and has the following hamiltonian $$ H = \sum_{i=1}^N \log{(1+n_i)} $$ I have to ...
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1answer
135 views

Physicists use of “Information”

Elsewhere here on physics.se I learned that Information contained in a physical system = the number of yes/no questions you need to get answered to fully specify the system. That is, however, ...
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1answer
65 views

Physical significance of free energy in canonical ensemble?

From macroscopic thermodynamics I understand that the free energy equals the total energy of the system minus the energy it would have cost a thermal reservoir to create it. So any energy in "excess" ...
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1answer
70 views

Changing Summation to Integral

This is the text from Reif Statistical mechanics. In the screenshot he changes the summation to integral(Eq. 1.5.17) by saying that they are approximately continuous values. However, I don't see how. ...
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The Borel Argument - 1 cm displacement of hydrogen atom on distant star effect on microstates on system on Earth

I was wondering if anyone had the derivation to Borel's famous arguement that a displacement of 1 cm on a particle on a distant star would significantly affect the microstate configuration on a system ...
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1answer
121 views

Why does many body localized phase follow Poisson distribution?

In many body localization (MBL) phenomenon, the ergodic phase follows a Gaussian distribution and MBL phase follows Poisson distribution. I can understand the Gaussian distribution, as this follows ...
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1answer
27 views

Simulate cluster in a heat bath

I want to simulate a cluster in a heat bath. I always thought I had to simulate it first in the microcanonical ensemble (N,V,E) until I reach the equilibrium. In the equilibrium the temperature will ...
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1answer
62 views

How to calculate the logarithm of an outer product?

Im tryting to calculate the entanglement entropy of a subsystem: $$S_\text{ent}=-k\cdot \mathrm{Tr}(\rho_A\log(\rho_A))$$ Let $$ \rho_A=\vert\uparrow\rangle\langle\uparrow\vert$$ from what I can ...
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1answer
37 views

One-dimensional polymer (Gibbs canonical ensemble)

Let's consider a polymer that is formed by an horizontal linear chain of $N$ disc-shaped monomers. Each monomer can either adopt either a vertical alignment (with length $l_1$ and energy $E_1$) or an ...
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2answers
137 views

How is spin-state of an electron defined?

An electron possesses ‘spin’ even at absolute zero. But how are the spin-states identified? It couldn’t be in terms of ‘energy’ states because well, the electron does not possess any energy at all.
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1answer
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Einstein solid: one or three dimensional quantum harmonic oscillator?

The Einstein model for solids assumes all atoms vibrate with the same frequency $\omega$, each atom being modeled as a quantum harmonic oscillator. The thing is: solids are three-dimensional objects, ...
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1answer
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Grand canonical ensemble and chemical potential $\mu=0$

In the grand canonical ensemble a system can exchange particles with a reservoir so its number of particles is not fixed. So what does it mean that $\mu=0$ implies that the number $N$ of particles is ...
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Minimisation of grand free potential at equilibrium?

I have a couple of questions regarding the grand potential at equilibrium. I am well aware that in the grand canonical ensemble, particles and energy can be exchanged. I also understand why the grand ...
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1answer
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Could micro-states not have equal probability as assumed? [closed]

How could micro-states have uniform distribution in a 3D space unless there was 0 Gravity? Gravity violates the assumption of equal probability as it makes matter move in a predictable direction. ...
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3answers
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Why can we ignore the interaction between heat reservior and our system in canonical ensemble?

When we derive the expression of canonical ensemble, we write the Hamiltonian of the whole system which contains our system and a heat reservior in the form, $$ H_{\text{whole}} = H_1(\boldsymbol x) + ...
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1answer
120 views

Alternate interpretation of number of degenerate fermions formula in phase space

I'm writing personnal notes on statistical mechanics and I'm tempted to write something that may turn out to be false. So I need a confirmation/infirmation and opinions on the following (I suspect it'...
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1answer
39 views

Proof of equipartition theorem for $i$ not equal to $j$

The generalized equipartition theorem (where variables need not be quadratic) states that if $x_i$ is a canonical variable (position or momentum variable), then $$\left\langle x_i \frac{∂H}{∂x_j}\...
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1answer
184 views

Ising model as quantum model?

I've read in a few papers things that use the fact that the $2D$ Ising model can be interpreted as a $1+1$ quantum spin model. I haven't been able to find this description and would like to read about ...
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Converting a discrete statistical energy distribution to a continuous version

The probability of finding a particle at a particular energy level when energy is considered discrete is according to Boltzmann: $$P(E_j) = \frac{g_j\cdot e^{-\beta E_j}}{\sum_{j=1}^\infty g_j \cdot ...
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2answers
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Increase of entropy as statistical necessity via fundamental assumption of statistical mechanics

My statistical physics books reasons that the increase of entropy for a closed system arises naturally from statistics. Outline: Fundamental Assumption of Statistical Mechanics: For a system at ...
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1answer
124 views

In statistical mechanics, why do we consider number of states of a system in energy interval?

In statistical mechanics,when we go for calculating the no. of accessible micro states of a system, I notice that we always calculate the no. of micro states of that system in some energy interval say ...
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5answers
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Physics of a burning log of firewood

According to my knowledge, heat is nothing but the result of the vibrations of atoms and molecules. I guess this mean that in heating up a gas or liquid, we are increasing the rate at which the ...
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1answer
39 views

Are quasistatic processes in thermodynamics achievable or an approximation?

According to Wikipedia: "In thermodynamics, a quasistatic process is a thermodynamic process that happens slowly enough for the system to remain in internal equilibrium." But if we look things more ...
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electron free mean path understanding

I am having difficulties to grasp the idea that the probability a particle is absorbed between $x$ and $x + dx$ is given by $$dP(x) = \frac{I(x) - I(x+dx)}{I_0} = \frac{1}{l} e^{-\frac{x}{l}}dx$$ ...
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Mean field theory of Potts model (equation solution)

When considering a $q$ states Potts model in mean field approximation, one finds the following free energy: $$ \beta f(s) = \frac{1+(q-1)s}{q}\log{\left[ 1 + (q-1)s \right]} + \frac{(q-1)(1-s)}{q}\log{...
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1answer
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Equivalence of thermodynamic ensembles

It is often argued that thermodynamic ensembles are equivalent in the sense that no matter what ensemble one uses for the calculations, one should end up in the same macroscopic equations of state. ...
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1answer
39 views

Confusion about microstate probabilities

So I know we have the Gibbs Entropy Formula $$S=k\sum_i p_i \ln p_i$$ And I've seen the probabilities for the microstates be said to be the Boltzmann probabilities $$p_i\propto e^{-\frac{E_i}{kT}}$$ ...
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2answers
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What is role of nucleation centre in the formation of ice?

Water must be impure so that the impurities can act as nucleation centre for ice to form. What is role of nucleation centre? Why can not ice form with some nucleation centre? About existing (and ...
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1answer
76 views

Do Maxwell relations hold during a phase transition?

Maxwell relations are found by taking mixed derivatives of a thermodynamic potential. Does this mean that they do not hold at a first-order phase transition, where the thermodynamic potential is ...
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1answer
42 views

Is there an analogue to Fermi-Dirac statistics for interacting electrons?

Alongside the formula I've seen for Fermi-Dirac statistics: $$\frac{1}{e^{\beta(E-\mu)}+1}=f(E)$$ I often see the addendum that this is only true for non-interacting electrons. By that, I sort of ...
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1answer
137 views

Thermodynamic Beta and Inverse Temperature

Following on from my previous question: Exponential form of Boltzmann Distribution I am now trying to understand the relationship between the thermodynamic beta and the inverse temperature. ...
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1answer
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Are statistical fields commutative?

In both statistical field theory and quantum field theory one computes average values / time ordered expectation values of functionals of fields with the path integral. I have two related questions: ...
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What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?

This question was listed as one of the questions in the proposal (see here), and I didn't know the answer. I don't know the ethics on blatantly stealing such a question, so if it should be deleted or ...
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2answers
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Why are degenerate states more likely to be filled at a given temperature?

Consider if we have a simple two-level toy model, where the ground state has energy $E_0 = 0$ and the excited state has energy $E_1 = \epsilon$ and degeneracy $g$. The partition function for this ...
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2answers
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Deriving the ideal gas law from relative entropy, instead of differential entropy, to avoid negative entropy

Deriving the ideal gas law is one of the first examples presented in introductory statistical physics. It is derived using the differential entropy. Let $\chi$ be an uncountable set. Then the ...
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1answer
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Correlation function of XY-model with vortices

The action for XY-model without magnetic field (in continuum limit) is $$S=\int d^2x\,(\partial_{\mu}\phi)^2,$$ which gives the following motion equation: $$\Delta\phi=0.$$ Single-valued, smooth ...